AN ALMOST COMPLEX CASTELNUOVO DE FRANCHIS THEOREM INDRANIL BISWAS AND MAHAN MJ Abstract. Given a compact almost complex manifold, we prove a Castelnuovo–de Franchis type theo- rem for it. 1. Introduction Given a smooth complex projective variety X, the classical Castelnuovo-de Franchis theorem [2, 4] 0 1 associates to an isotropic subspace of H (X, ΩX ) of dimension greater than one an irrational pencil on X. The topological nature of this theorem was brought out by Catanese [3], who established a bijective correspondence between subspaces U H1(X, C) of the form U U with U being a maximal isotropic subspace of H1(X, C) of dimension g ⊂ 1 and irrational fibrations⊕ on X of genus g. The purpose of this e note is to emphasize this topological content≥ further. We extract topological hypotheses that allow the theorem to go through when we have only an almost complex structure. In what follows, M will be a compact smooth manifold of dimension 2k. Let J : TM TM M −→ be an almost complex structure on M, meaning J J = Id . M ◦ M − T M Definition 1.1. We shall say that a collection of closed complex 1–forms ω1, ,ωn on M are in general position if ··· (1) the zero-sets Z(ω ) := x M ω (x)=0 M i { ∈ | i } ⊂ are smooth embedded submanifolds, and (2) these submanifolds Z(ωi) intersect transversally. We are now in a position to state the main theorem of this note. Theorem 1.2. Let M be a compact smooth 2k–manifold equipped with an almost complex structure JM . Let ω , ,ω be closed complex 1–forms on M linearly independent over C, with g 2, such that 1 ··· g ≥ each ωi is of type (1 , 0), meaning ωi(JM (v)) = √ 1 ωi(v) for all v TM, and • ω are in general position with ω ω = 0 for all−i , j·. ∈ • i i ∧ j arXiv:1502.06205v2 [math.CV] 13 Apr 2016 Then there exists a smooth almost holomorphic map f : M C to a compact Riemann surface of −→ genus at least g, and there are linearly independent holomorphic 1–forms η1 , , ηg on C, such that ∗ ··· ωi = f ηi for all i. 2. Leaf space and almost complex blow-up 2.1. Leaf space. Assume that ω are forms as in Theorem 1.2. Since ω ω = 0 for all 1 i , j g, i i ∧ j ≤ ≤ it follows that there are complex valued smooth functions fi,j such that ωi = fi,j ωj (1) wherever ω = 0. Hence the collection j 6 = ω , ,ω W { 1 ··· g} 2000 Mathematics Subject Classification. 32Q60, 14F45. Key words and phrases. Almost complex structure, Castelnuovo–de Franchis theorem, Riemann surface, fundamental group. 1 2 INDRANIL BISWAS AND MAHAN MJ determines a complex line subbundle of the complexified cotangent bundle (T ∗M) C over the open subset ⊗ g V := M Z(ω ) M. \ i ⊂ i\=1 Lemma 2.1. Let := v T V ωi(v)=0 1 i g T V be the distribution on V defined by . Then is integrableF { and∈ defines| a foliation∀ of real≤ codimension≤ } ⊂ two on V . W F Proof. For any x M and 1 i g, consider the R–linear homomorphism ∈ ≤ ≤ ωx : T M C , v ω (x)(v) . i x −→ 7−→ i x x x Since ωi is of type (1 , 0), if ωi(x) = 0, then ωi is surjective. Also, ωj is a scalar multiple of ωi because ω ω = 0. Therefore, we conclude6 that the distribution on V is of real codimension two. i ∧ j F Since dw = 0 for all i, it follows that the distribution is integrable. i F Remark 2.2. Since Z(ωi) are smooth embedded submanifolds by hypothesis, and ωi are of type (1 , 0), it follows that Z(ωi) are almost complex submanifolds of M, meaning JM preserves the tangent subbun- dle TZ(ωi) (TM) Z(ωi). Consequently, all the intersections of the Z(ωi)’s are also almost complex submanifolds⊂ and are| therefore even dimensional. Clearly, each Z(ωi) has complex codimension at least one (real codimension at least two) as an almost complex submanifold. Therefore, the complement M V has complex codimension at least two. \ 2.2. Almost complex blow-up. We shall need an appropriate notion of blow-up in our context. Sup- pose K M is a smooth embedded submanifold of M of dimension 2j such that JM (TK) = TK. By ⊂ g Remark 2.2, the submanifold i=1 Z(ωi) satisfies these conditions. Note that JM induces an automor- phism T J : ((TM) )/TK ((TM) )/TK M/K |K −→ |K of the quotient bundle over K. Since JM is an almost complex structure it follows that JM/K JM/K = Id . Therefore, ((TM) )/TK is a complex vector bundle on K of rank k j.◦ We would − ((T M)|K )/T K |K − like to replace K by the (complex) projectivized normal bundle P(((TM) K )/TK) which will be called the almost complex blow-up of M along K. | For notational convenience, the intersection g Z(ω ) will be denoted by . i=1 i Z We first projectivize (TM) to get a CPk−1Tbundle over ; this CPk−1 bundle will be denoted by . |Z Z B So parametrizes the space of all (real) two dimensional subspaces of (TM) Z preserved by JM (such twoB dimensional subspaces are precisely the complex lines in (TM) equipped| with the complex vector |Z bundle structure defined by JM ). Let π : B −→ Z be the natural projection. For notational convenience, the pulled back vector bundle π∗((TM) ) will be denoted by π∗TM. |Z Let T := kernel(dπ) T π ⊂ B be the relative tangent bundle, where dπ : T π∗T is the differential of π. The map π is almost B −→ Z holomorphic, and Tπ has the structure of a complex vector bundle. It is known that the complex vector bundle T is identified with the vector bundle HomC( , (π∗TM)/ ) = ((π∗TM)/ ) C ∗, where π L L L ⊗ L π∗TM L ⊂ is the tautological real vector bundle of rank two; note that both and (π∗TM)/ have structures of L L complex vector bundles given by JM , and the above tensor product and homomorphisms are both over C. Therefore, the pullback π∗TM splits as ((π∗TM)/ ) = (T ) . L⊕ L L⊕ π ⊗ L The image of the zero section of the complex line bundle is identified with , and the normal L −→ B B bundle of is identified with . A small deleted normal neighborhood UB of in can be identified with a deletedB ⊂ neighborhoodL U of L in M. Let U := U be the neighborhoodB L of in (it is no Z B B B B L S ANALMOSTCOMPLEXCASTELNUOVODEFRANCHISTHEOREM 3 longer a deleted neighborhood), where is again identified with the image of the zero section of . In B L the disjoint union (M ) U B, we may identify U with UB. The resulting topological space will be called the almost complex\ Z ⊔ blow-up of M along . Z Remark 2.3. (1) Consider the foliation on the deleted neighborhood U of in M defined by . Let LU denote the leaf space for it. After identifying with the image ofZ the zero section ofFπ∗TM, we obtain B locally, from a neighborhood of , a map to the leaf space LU . (2) Note that the notion of an almostB complex blow-up above is well-defined up to the choice of an identification of U with UB. Therefore, the construction yields a well-defined almost complex manifold, up to an isomorphism. 3. Proof of Theorem 1.2 We will first show that the leaves of in Section 2 are proper embedded submanifolds of V = M . F \ Z Suppose some leaf L0 does not satisfy the above property. Then there exists x V and a neighborhood U of x such that U L contains infinitely many leaves (of ) accumulating at∈ x. Let σ be a smooth x x 0 F path starting at x transverseT to . Thus for every ǫ > 0 there exist F (1) distinct (local) leaves F , F U , such that (globally) F , F L , 1 2 ⊂ x F 1 2 ⊂ 0 (2) points y F , j = 1 , 2, T j ∈ j (3) a sub-path σ12 σ Ux joining y1 and y2, and (4) a path τ L⊂joining⊂ y and y such that 12 ⊂ 0 1 2 ωi < ǫ i . | Zσ12 | ∀ Since is in the kernel of each ω , F i ωi = 0, i . | Zτ12 | ∀ Setting ǫ smaller than the absolute value of any non-zero period of the ωi’s, it follows that ωi =0 1 i g . Zτ12∪σ12 ≤ ≤ Hence ωi = 0 1 i g . Zσ12 ∀ ≤ ≤ ′ Taking limits we obtain that ωi(σ (0)) = 0; but this contradicts the choice that σ is transverse to . Therefore, the leaves of are proper embedded submanifolds of V . F F Consequently, the leaf space D = V/ hFi for the foliation is a smooth 2-manifold. Let F q : V D −→ be the quotient map. Remark 3.1. The referee kindly pointed out to us the following considerably simpler proof of the fact that leaves of are closed in V : F By equation (1), we have ωi = fi,j ωj. Since ωi are closed, d(f ) ω = 0, i , j . i,j ∧ j ∀ 1 The functions fi,j define a map f : V CP . Since d(fi,j ) ωj = 0, it follows that f is constant on the leaves of the foliation defined by−→ the forms ω , 1 ileqn∧.